And obviously a plane is far more likely to lose lift and stall the higher you fly, as the air thins and lift is reduced.
Nope. Lift is proportional to the square of the air speed and will occur at the same IAS no matter what the density of the local atmosphere. It is the TAS that will change with height - and not many aircraft I know fly on TAS.
It is lack of thrust that is the snag at high level.
Mog